Understanding the Composition of Functions and Domain

Oct 13, 2025 by miodeya.com 122 views

Understanding the Composition of Functions and Domain

In mathematics, the composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. Given two functions, f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x), is defined as f(g(x)). This means that we first apply the function g to the input x, and then apply the function f to the result.

In this article, we will explore the composition of two functions, f(x) and g(x), and determine the domain of the resulting function (f ∘ g)(x). We will use the given functions f(x) = (x-3)/(x) and g(x) = 5x - 4 to illustrate the concept.

The Composition of Functions

To find the composition of f and g, we need to substitute g(x) into f(x) in place of x. This gives us:

(f ∘ g)(x) = f(g(x)) = f(5x - 4)

Now, we substitute g(x) = 5x - 4 into f(x) = (x-3)/(x):

(f ∘ g)(x) = f(5x - 4) = ((5x - 4) - 3) / (5x - 4)

Simplifying the expression, we get:

(f ∘ g)(x) = (5x - 7) / (5x - 4)

Determining the Domain of the Composition

The domain of a function is the set of all possible input values for which the function is defined. In the case of the composition (f ∘ g)(x), the domain is determined by the restrictions on the input values of both functions f and g.

For the function f(x) = (x-3)/(x), the denominator cannot be zero, so we must exclude x = 0 from the domain. Additionally, the function g(x) = 5x - 4 is defined for all real numbers, so there are no restrictions on the input values of g.

However, when we substitute g(x) into f(x), we get a new function (f ∘ g)(x) = (5x - 7) / (5x - 4). The denominator of this function cannot be zero, so we must exclude x = 4/5 from the domain.

Therefore, the domain of the composition (f ∘ g)(x) is all real numbers except x = 4/5.

Conclusion

In conclusion, the domain of the composition (f ∘ g)(x) is all real numbers except x = 4/5. This is because the denominator of the function (f ∘ g)(x) cannot be zero, and x = 4/5 would result in a zero denominator.

Answer

The correct answer is:

C. {x | x ≠ 4/5}

This answer reflects the domain of the composition (f ∘ g)(x), which is all real numbers except x = 4/5.

Final Thoughts

In this article, we explored the composition of two functions, f(x) and g(x), and determined the domain of the resulting function (f ∘ g)(x). We used the given functions f(x) = (x-3)/(x) and g(x) = 5x - 4 to illustrate the concept. The domain of the composition (f ∘ g)(x) is all real numbers except x = 4/5. This is an important concept in mathematics, and it has many practical applications in fields such as physics, engineering, and computer science.
Q&A: Composition of Functions and Domain

In our previous article, we explored the composition of two functions, f(x) and g(x), and determined the domain of the resulting function (f ∘ g)(x). In this article, we will answer some frequently asked questions about the composition of functions and domain.

Q: What is the composition of functions?

A: The composition of functions is a way of combining two or more functions to create a new function. Given two functions, f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x), is defined as f(g(x)). This means that we first apply the function g to the input x, and then apply the function f to the result.

Q: How do I find the composition of two functions?

A: To find the composition of two functions, f(x) and g(x), you need to substitute g(x) into f(x) in place of x. This gives you:

(f ∘ g)(x) = f(g(x)) = f(5x - 4)

Now, you substitute g(x) = 5x - 4 into f(x) = (x-3)/(x):

(f ∘ g)(x) = f(5x - 4) = ((5x - 4) - 3) / (5x - 4)

Simplifying the expression, you get:

(f ∘ g)(x) = (5x - 7) / (5x - 4)

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In the case of the composition (f ∘ g)(x), the domain is determined by the restrictions on the input values of both functions f and g.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to identify any restrictions on the input values. For example, if the function has a denominator that cannot be zero, you need to exclude that value from the domain.

Q: What is the difference between the domain and the range of a function?

A: The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For example, if a function has a domain of all real numbers, but only produces positive output values, then the range is the set of all positive real numbers.

Q: Can I have a function with an empty domain?

A: Yes, it is possible to have a function with an empty domain. This occurs when the function is undefined for all possible input values. For example, the function f(x) = 1/x has an empty domain because it is undefined when x = 0.

Q: Can I have a function with an infinite domain?

A: Yes, it is possible to have a function with an infinite domain. This occurs when the function is defined for all possible input values. For example, the function f(x) = x^2 has an infinite domain because it is defined for all real numbers.

Q: How do I graph a composition of functions?

A: To graph a composition of functions, you need to first graph the individual functions f(x) and g(x). Then, you can use the graph of g(x) to determine the input values for f(x). Finally, you can graph the composition (f ∘ g)(x) by using the output values of g(x) as the input values for f(x).

Q: Can I have a composition of functions with multiple inputs?

A: Yes, it is possible to have a composition of functions with multiple inputs. This occurs when the function g(x) has multiple input values, and the function f(x) takes those input values as input. For example, the function f(x, y) = x^2 + y^2 is a composition of two functions, f(x) = x^2 and g(y) = y^2.

Q: Can I have a composition of functions with multiple outputs?

A: Yes, it is possible to have a composition of functions with multiple outputs. This occurs when the function f(x) has multiple output values, and the function g(x) takes those output values as input. For example, the function f(x) = (x, x^2) is a composition of two functions, f(x) = x and g(x) = x^2.

Conclusion

In conclusion, the composition of functions is a powerful tool for combining two or more functions to create a new function. By understanding the domain and range of a function, you can determine the input and output values of the composition. By graphing the composition of functions, you can visualize the relationship between the input and output values. By exploring the composition of functions with multiple inputs and outputs, you can create complex and interesting functions.